Traversal times for random walks on small-world networks

Effective-medium approximation for lattice random walks with long-range jumps

We consider the random walk on a lattice with random transition rates and arbitrarily long-range jumps. We employ Bruggeman's effective-medium approximation (EMA) to find the disorder-averaged (coarse-grained) dynamics. The EMA procedure replaces the disordered system with a cleverly guessed reference system in a self-consistent manner. We give necessary conditions on the reference system and discuss possible physical mechanisms of anomalous diffusion. In the case of a power-law scaling between transition rates and distance, lattice variants of Lévy-flights emerge as the effective medium, and the problem is solved analytically, bearing the effective anomalous diffusivity. Finally, we discuss several example distributions and demonstrate very good agreement with numerical simulations.

An alternative approach to characterize the topology of complex networks and its application in epidemic spreading

Based on the mean-field approach, epidemic spreading has been well studied. However, the mean-field approach cannot show the detailed contagion process, which is important in the control of epidemic. To fill this gap, we present a novel approach to study how the topological structure of complex network influences the concrete process of epidemic spreading. After transforming the network structure into hierarchical layers, we introduce a set of new parameters, i.e., the average fractions of degree for outgoing, ingoing, and remaining in the same layer, to describe the infection process. We find that this set of parameters are closely related to the degree distribution and the clustering coefficient but are more convenient than them in describing the process of epidemic spreading. Moreover, we find that the networks with exponential distribution have slower spreading speed than the networks with power-law degree distribution. Numerical simulations have confirmed the theoretical predictions.

Extensions of effective-medium theory of transport in disordered systems

The effective-medium theory of transport in disordered systems, whose basis is the replacement of spatial disorder by temporal memory, is extended in several practical directions. Restricting attention to a one-dimensional system with bond disorder for specificity, a transformation procedure is developed to deduce explicit expressions for the memory functions from given distribution functions characterizing the system disorder. It is shown how to use the memory functions in the Laplace domain forms in which they first appear, and in the time domain forms which are obtained via numerical inversion algorithms, to address time evolution of the system beyond the asymptotic domain of large times normally treated. An analytic but approximate procedure is provided to obtain the memories, in addition to the inversion algorithm. Good agreement of effective-medium theory predictions with numerically computed exact results is found for all time ranges for the distributions used except near the percolation limit, as expected. The use of ensemble averages is studied for normal as well as correlation observables. The effect of size on effective-medium theory is explored and it is shown that, even in the asymptotic limit, finite-size corrections develop to the well-known harmonic mean prescription for finding the effective rate. A percolation threshold is shown to arise even in one dimension for finite (but not infinite) systems at a concentration of broken bonds related to the system size. Spatially long-range transfer rates are shown to emerge naturally as a consequence of the replacement of spatial disorder by temporal memories, in spite of the fact that the original rates possess nearest neighbor character. Pausing time distributions in continuous-time random walks corresponding to the effective-medium memories are calculated.

Random-walk access times on partially disordered complex networks: an effective medium theory

An analytic effective medium theory is constructed to study the mean access times for random walks on hybrid disordered structures formed by embedding complex networks into regular lattices, considering transition rates F that are different for steps across lattice bonds from the rates f across network shortcuts. The theory is developed for structures with arbitrary shortcut distributions and applied to a class of partially disordered traversal enhanced networks in which shortcuts of fixed length are distributed randomly with finite probability. Numerical simulations are found to be in excellent agreement with predictions of the effective medium theory on all aspects addressed by the latter. Access times for random walks on these partially disordered structures are compared to those on small-world networks, which on average appear to provide the most effective means of decreasing access times uniformly across the network.

Efficiency of trapping processes in regular and disordered networks

We use a Markov method to study the efficiency of trapping processes involving both a random walker and a deep trap in regular and disordered networks. The efficiency is gauged by the mean absorption time (average of the mean number of steps performed by the random walker before being absorbed by the trap). We compute this quantity in terms of different control parameters, namely, the length of the walker jumps, the mobility of the trap, and the degree of spatial disorder of the network. For a proper choice of the system size, we find in all cases a nonmonotonic behavior of the efficiency in terms of the corresponding control parameter. We thus arrive at the conclusion that, despite the decrease of the effective system size underlying the increase of the control parameter, the efficiency is reduced as a result of an increase of the escape probability of the walker once it finds itself in the interaction zone of the trap. This somewhat anti-intuitive effect is very robust in the sense that it is observed regardless of the specific choice of the control parameter. For the case of a ring lattice, results for the mean absorption time in systems of arbitrary size are given in terms of a two-parameter scaling function. For the case of a mobile trap, we deal with both trapping via a single channel (walker-trap overlap) and via two channels (walker-trap overlap and walker-trap crossing), thereby generalizing previous work. As for the disordered case, our analysis concerns small world networks, for which we see several crossovers of the absorption time as a function of the control parameter and the system size. The methodology used may be well suited to exploring characteristic time scales of encounter-controlled phenomena in networks with a few interacting elements and the effect of geometric constraints in nanoscale systems with a very small number of particles.

The application of graph theoretical analysis to complex networks in the brain

Considering the brain as a complex network of interacting dynamical systems offers new insights into higher level brain processes such as memory, planning, and abstract reasoning as well as various types of brain pathophysiology. This viewpoint provides the opportunity to apply new insights in network sciences, such as the discovery of small world and scale free networks, to data on anatomical and functional connectivity in the brain. In this review we start with some background knowledge on the history and recent advances in network theories in general. We emphasize the correlation between the structural properties of networks and the dynamics of these networks. We subsequently demonstrate through evidence from computational studies, in vivo experiments, and functional MRI, EEG and MEG studies in humans, that both the functional and anatomical connectivity of the healthy brain have many features of a small world network, but only to a limited extent of a scale free network. The small world structure of neural networks is hypothesized to reflect an optimal configuration associated with rapid synchronization and information transfer, minimal wiring costs, resilience to certain types of damage, as well as a balance between local processing and global integration. Eventually, we review the current knowledge on the effects of focal and diffuse brain disease on neural network characteristics, and demonstrate increasing evidence that both cognitive and psychiatric disturbances, as well as risk of epileptic seizures, are correlated with (changes in) functional network architectural features.

Graph theoretical analysis of complex networks in the brain

Since the discovery of small-world and scale-free networks the study of complex systems from a network perspective has taken an enormous flight. In recent years many important properties of complex networks have been delineated. In particular, significant progress has been made in understanding the relationship between the structural properties of networks and the nature of dynamics taking place on these networks. For instance, the 'synchronizability' of complex networks of coupled oscillators can be determined by graph spectral analysis. These developments in the theory of complex networks have inspired new applications in the field of neuroscience. Graph analysis has been used in the study of models of neural networks, anatomical connectivity, and functional connectivity based upon fMRI, EEG and MEG. These studies suggest that the human brain can be modelled as a complex network, and may have a small-world structure both at the level of anatomical as well as functional connectivity. This small-world structure is hypothesized to reflect an optimal situation associated with rapid synchronization and information transfer, minimal wiring costs, as well as a balance between local processing and global integration. The topological structure of functional networks is probably restrained by genetic and anatomical factors, but can be modified during tasks. There is also increasing evidence that various types of brain disease such as Alzheimer's disease, schizophrenia, brain tumours and epilepsy may be associated with deviations of the functional network topology from the optimal small-world pattern.

Synchronization in weighted uncorrelated complex networks in a noisy environment: optimization and connections with transport efficiency

Motivated by synchronization problems in noisy environments, we study the Edwards-Wilkinson process on weighted uncorrelated scale-free networks. We consider a specific form of the weights, where the strength (and the associated cost) of a link is proportional to (kikj)beta with ki and kj being the degrees of the nodes connected by the link. Subject to the constraint that the total edge cost is fixed, we find that in the mean-field approximation on uncorrelated scale-free graphs, synchronization is optimal at beta*= -1 . Numerical results, based on exact numerical diagonalization of the corresponding network Laplacian, confirm the mean-field results, with small corrections to the optimal value of beta*. Employing our recent connections between the Edwards-Wilkinson process and resistor networks, and some well-known connections between random walks and resistor networks, we also pursue a naturally related problem of optimizing performance in queue-limited communication networks utilizing local weighted routing schemes.